# Symbols and sets

Purpose

This unit builds upon the students’ experiences of making, naming and recognising common fractions using different physical representations. Its purpose is to develop understanding of fractions of sets, and the formal language and symbols associated with simple fractions and their representations.

Specific Learning Outcomes
• Read and write words and symbols for fractions.
• Introduce the terms ‘unit fraction’ and ‘proper fraction’, numerator and denominator.
• Make and understand different ways to represent 1 (whole).
• Use regional representations to find fractions of sets.
• Solve problems that involve finding the whole from a part.
• Find fractions of sets showing solutions in multiple ways including connecting fractions of sets with division.
Description of Mathematics

Proportional thinking has been introduced informally in the previous unit with a range of experiences that have involved making, naming and recognising common fractions using different physical representations. The focus of these experiences has been on developing understanding of equal parts and their names (the denominator) compared with ‘bits’ or unequal parts. Students have also experienced selecting a number of equal parts (the numerator), however this formal fractional language and the symbols associated with these fractions have yet to be introduced to the students. That is one purpose of these lessons.

Working with regional representations of fractions provides a sound basis for exploring fractions of sets. The equal partitioning of regions has built the fundamental understanding necessary to finding fractions of sets which demands an additional layer of complexity as students work with numbers of items in an equal part, and with associated number operations. If students are to achieve understanding here they must clearly know what is the whole and what is the unit of partition.

In applying the principles of the Teaching Model (Book 3 Getting Started p5) students use physical representations of fractions of regions as a ‘tool’ to support them as they find fractions of sets. The combination of these regional and set representations consolidate the conceptual understanding of fractions themselves and also provide a tangible reference as students rely less on materials, begin to image these and come to understand and use the relational number properties.

Students are more frequently posed problems that require them to find a fraction of an area, length or set. This involves them in partitioning or dividing a whole. The reciprocal experiences of beginning with a part (or unit) and adding or multiplying to find the whole must also be provided if deeper understanding is to be achieved.

These ideas are presented in five sessions however, as they include complex concepts, they can be extended over a longer period of time. A number of games are included. Whilst these are introduced and used within sessions to consolidate ideas, they can also be added to the class or group independent activities.

#### Links to the Number Framework

Stages 5- 6

This unit supports teaching and learning activities in the Student Fractions e-ako 3 and 4 and complements the learning activities in Book 7 Teaching Fractions, Decimals and Percentages.

Required Resource Materials
• Play dough
• Plastic knives
• Paper shapes
• Scissors
• Sets of Fraction Circles or similar materials (Material Master 4-19)
• Dice
• Plastic beans
Activity

#### Session 1

The purpose of this session is to learn how to make and describe equal parts and to read and write words and symbols for fractions. The formal language of ‘unit fraction’ and ‘proper fraction’ is also introduced.

Activity 1

Begin by placing a length of play dough and a knife in a place where all students in the group (class) can see. Distribute to each pairs of students word cards (only) from Attachment 1.
In their pairs have them discuss how they would make the equal part on their word card, if they were to use the play dough. Chose several students to explain this to the group. Listen for and highlight a description of the number of cuts that will be made and the language of equal parts.
Have a student demonstrate with materials if appropriate.

Activity 2

1. Randomly distribute symbol cards (Attachment 1) to the student pairs. Have students discuss the symbol and if possible agree about how to read it.

2. Pairs display and read their word cards aloud to the group, one at a time and as they do so, the pair that has the matching symbol card offers it to the holders of the word card. The donor pair must explain why they are giving the symbol to the word holder pair. Praise those who give clear explanations of the symbolic representation of the fractional part.
Highlight the fact that each of the symbols is known as a unit fraction because it has 1 as the top number and it tells that there is just 1 of the equal parts being referred to. (If you have more than 10 student pairs there will be some duplication of fractional pieces. This should be managed appropriately.)
NB Highlight the fact that one quarter and one fourth are different names for the same part. Some students will know some or all of the symbols and others may know none. Encourage student lead, rather than teacher lead explanations.

Activity 3

1. Make available to the students a range of shapes of coloured paper.
Explain that they are to select a paper shape each and fold it to make a fractional part matching their word and symbol, then write the word and symbol on each of the equal parts. They should be encouraged first to consider which shape would be the best to choose, given their fraction.
For example those with third, fifth, seventh, and ninth parts could be guided to choose a paper strip as it is easier to work with.
Have the students select a shape and complete the task discussing folds with their partner as appropriate. This may be challenging for those with fractions with odd numbered denominators. Encourage those who finish quickly (for example those with 1/2 or 1/4) to complete the task with a different shape, or, as appropriate, to try a different fraction.

2. Have students pair share their results, then talk as a class/group about why some fractions were easier to make than others and how students approached the ‘trickier ‘ones. (even numbered fractional parts can be folded and folded again).

3. Have all students make one tenth with a paper strip and have them write 1/10 and the word one tenth on each of the equal pieces.

Activity 4

1. Ensure that the students have writing materials and scissors available. Have each student cut their tenth strip into separate tenth pieces. Have them count their tenth pieces together: one tenth, two tenths, … ten tenths.

2. Write ‘unit fraction’ on the class/group chart/. Repeat and write the explanation that any fraction with a top number of 1 is called a unit fraction because it is a single piece. Add the words ‘proper fraction’ to the group chart, explaining and record that a fraction in which the top number is smaller than the bottom number is called a ‘proper fraction’.

3. Write 3 on the class chart. Ask students to take 3 of their tenth pieces and write the words and symbol for these three parts. Discuss and model the fraction symbol as appropriate, highlighting the fact that the bottom number tells us how many parts altogether (ten) and the top number (3) tells us how many of those equal parts we have chosen.
Also point out that often we see the flat line separating these numbers (this line is called the vinculum) shown with a sloping line like this, 1/2 , 1/4 This is just a slightly different way of writing the fraction number.

4. Write other numbers of the chart, having students take that many tenths and recording the appropriate words and proper fraction.

Activity 5

Conclude this session by writing the words and symbols for common unit fractions and some other proper fractions. Brainstorm on the class chart/book what has been learned about fraction symbols.

#### Session 2

The purpose of this session is to introduce the language of numerator and denominator and practice using and interpreting fraction symbols as students work from whole to part and part to whole. An understanding of different ways of showing one whole is also developed.

Activity 1

1. Reread together the learning from the end of Session 1.

2. Distribute the cards from Attachment 1 and have each student read out their card. As they listen they should identify the person with a matching card. This person becomes is their partner for this session.

3. Ask what the names of the kinds of fractions that were learned in the last session (unit and proper fraction). Introduce a Fraction Dictionary. Have the students help you write these words with their definitions into the dictionary.

4. Write ‘numerator’ in the dictionary. Explain that it is the top number in a fraction. Have several students come up and write their favourite fraction and circle in a different colour the top number.
Ask student pairs to discuss what the job of the top number is and to suggest a definition of ‘numerator’. Record the best of these.
Ask if students know what the bottom number is called. Accept all suggestions then write denominator. Read the word together, and record the best student suggestion for a denominator definition.
Have several students again write their favourite fractions, this time writing over the denominator number in a different colour (not the same as that used for the numerator).

Activity 2

1. Make sets of fractions circles available to the students.
Pose and write on a chart the question: Can the numerator in a fraction be the same as the denominator?
Have student pairs discuss this then collect a Yes or No card (Attachment 3) from two piles at the front of the room. Have two pairs combine (if possible with a pair that has the opposite card) and discuss why/why not and show each other with fraction circle pieces.
If all agree they can chose one pair to explain their thinking to the class.

2. Write several examples 4/4, 3/3, 2/2 etc. and have students draw in the class/group book what this would look like. Ask students to say what whole number these fractions are equivalent to (1).

Activity 3

Have students play Roll for 3 in pairs.
(Purpose: to make 1 from equal parts and recognise the equivalent fraction notation for this.)
Student pairs share a fraction pieces page (see Material Master 4-19). Each student takes 3 different circles.

For example, Player one takes, 1/3, 1/8 and 1/4 and Player two takes, 1/2, 1/6, 1/5. The winner is the player who is first to complete their circles and has correctly recorded each one as a fraction: 3/3, 8/8 etc. once the circle is complete.
The players take turns to roll the dice and colour in that many parts of one of the circles they have selected.
The important rule is that they can colour fewer parts and keep building to make one whole, but they must, at some point, roll the exact number needed to complete a whole.
For example, Player One rolls 6. She colours 6/8 of her circle divided into eighths.  On her next turn she rolls 3. She cannot use this to complete her eighths circle because 6/8 + 3/8 is more than 8/8 (1 complete circle). She must roll 2 or two 1s in different rolls to compete 1 exactly. She can however work on her tenths circle and take 3/10 and add this to her 1/10.
Emphasise that the students should record their fraction additions.

This game can also be played with plastic fraction parts. Instead of colouring the sheet they will build their whole in front of them.

Activity 4

Ensure that each student has their own copy of both a Yes and No card (Attachment 3).
Introduce the Fraction Snap game. (Attachment 2).
Hold up selected pairs of cards from Fraction Snap asking the students to decide for themselves if the two cards match. If so they hold up a Yes card and if not a No card.
Discuss examples so the game is well understood. For example:  is a pair, and so is  All of the set of Fraction Snap cards is distributed evenly to up to four players who turn their own pile face down in front of them.
Each player takes a turn to turn up a card from their pile and place it face up in the centre of the group. As a student adds their card to the pile of face up cards, all players watch closely and are ready to say ‘Snap’, quickly putting their hand on the pile if the played card matches the one that was top of the face up pile. The first to say ‘Snap” collects the pile and adds it face down to the bottom of their existing pile.
The game continues until one person has all the cards or until players decide to stop.

#### Session 3

The purpose of this session is to use materials to reinforce the part to whole relationship and to use fractions of regions to build an understanding of fractions of sets. Children will make connections with equal sharing experiences in their own lives.

Connections between repeated addition and multiplication are made as part-to-whole fraction problems are explored.

Activity 1

1. Make recording material available to the students.
Distribute a shape from Attachment 4 to each student.

2. Explain that what they have is a fraction or part of a whole shape. They need to show with a drawing what the whole shape might look like. Model an example and show they can draw around their cardboard shape. (Alternatively attribute blocks or foam geometric shapes can be used. The students will need to be told what fraction of the whole they are working with.)

3. Give the students the opportunity to explore the problem before prompting with questions like, “If that is 1/4 of the shape, how many of pieces like that will be in the whole shape?” or, “Do you think that there might be another way to show the whole shape?”

4. Have students complete their drawings, writing the unit fraction in each part and an equivalent fraction for 1 beside their drawing (eg. 4/4, 2/2). Have them buddy share their results. Challenge the students to see how many ‘1’ shapes they can make for any single fraction piece.
For example if this is 1/4, the whole shape might look like any of these:   Have students repeat this with several more shapes, writing the fraction in each part and the one whole fraction (4/4, 8/8 etc.) beside the whole shapes.

Activity 2

1. Make plastic beans available to the students.
Have the students each take up to 4 beans and place them on their coloured fraction piece. 2. Explain that these beans, like their shape are just a fraction of a set of beans. It’s the same fraction as the fractions shape they have (1/4, 1/3, 1/2 etc.). Pose the question:
If this is a fraction of the set, how many beans are in the whole set?

3. Give the students the opportunity to explore the problem before prompting with questions like, “How quarters are in a whole set?” “How can you use your shape pictures to help you work out how many beans would be in the whole set?” If the students exploration is unsuccessful, stop the class/group and model an example, by putting the same number of beans on each of the fraction parts in the drawing of the whole shape and skip counting (or if appropriate multiplying) to reach a total.

4. Model and record several examples on the class/group chart.
For example : 1/4 of a whole set is 3 beans, 4/4 make 1 whole, so 4 lots of 3 beans will make 1 whole set. “3, 6, 9, 12” or “4 x 3 = 12”

5. Have students explore and record at least 3 more examples using different fractions and shapes and different small amounts of beans.

Activity 3

1. Conclude this session by asking students to give examples of when someone has shared with them and they had received an equal part of a whole set of something.

2. In the class book record some of the students’ story examples: For example:
Mia’s friend Amy gave her 1/2 of her jellybeans. Mia had 5. How many did Amy have altogether before she shared?
Tony received 6 pretzel sticks from Tama who told him he’d given him 1/3. How many did Tama have to start with?
Encourage the students to picture these fractional amounts and what the whole amount might look like. Ask students to describe what they pictured in their minds.

#### Session 4

The purpose of this session is to use materials to reinforce the whole to part relationship and to continue to use fractions of regions to build an understanding of fractions of sets. The key connection is made operation of division, which involves breaking down an amount or set into a number of smaller sets with the same number in each.

Activity 1

Begin this session by reviewing Session 2, posing some fractional part to whole contextual problems.
For example: You were given 6 cherries. This was one third of the total in the bag. How many were in the bag to start with?
Encourage students to image the problem and solution, but if appropriate, have a student model with an appropriate drawing.
Repeat with several examples, highlighting repeated addition and multiplication as strategies for reaching a solution.

Activity 2

Have the students working in pairs with recording materials available.
Pose the problem: Here is a container of strawberries. If there are 12 in the container and you are sharing these between you how many do you each get? What is 1/2 of 12?
Show and write how you work out your share, using pictures words and symbols.
Pose several more examples with different numbers in the container: What is half of: 14, 20, 21, 25?
Give the students time to draw, record and write about their sharing. Explain that these will be shared with other students and displayed.

Activity 3

1. Have students share their work and comment on any examples where students use fractions of regions (a shape divided into halves) to support their calculations.

2. On the class chart write:

Half of twelve
12 shared between 2
12 ÷ 2
12/2
Also 12 halves

Take time to discuss the important connections between these ideas with the students as we often assume that students understand these.
Highlight:
• half of twelve is 12 shared with 2
• when we share 12 with two people we can write 12 ÷ 2 (= 6)
• ÷ is the sign for division. It looks a bit like a fraction itself
• 12/2 also means twelve shared between two (which is 6:quotient)

In sharing both 21 and 25 between them, the student pairs will have had to share 1 strawberry and will have written 1/2. Highlight that we read this as ‘half’ and that this symbol is also showing 1 ÷ 2, or 1 shared between 2.
The symbol is an expression of both the problem itself and the quotient.
These key ideas about mathematical notation should be regularly reviewed.

Activity 4

Remind students how they used shapes (regions) in Session 3 to help solve problems. They may find them useful now.
Now pose equal sharing word problems such as:
12 strawberries shared between 3 people, 1/3 of 12 (or 12/3)
16 shared between 4, or 1/4 of 16 (or 16/4)
20 ÷ 5, or 1/5 of 20 (or 20/5)
21 ÷ 4, or 1/4 of 21 (or 21/4)
Have students use pictures words and symbols to record their solutions to the problems. Have students pair share their work.

Activity 5

Pairs of students can be challenged to write their own fraction problems for their partner to solve.

#### Session 5

The purpose of this session is to develop a conceptual understanding of finding a more than a unit fraction of a set. The language which has been introduced in this unit is consolidated.

Activity 1

1. Begin this session by highlighting strengths of some of the student work from Session 4, noticing the way they have drawn their diagrams and recorded their ideas using words and symbols.

2. Make coloured beans and paper available to pairs of students.
On the class chart write and pose this problem:
There are 25 beans in a packet. Each row that you plant uses 2/5 of a packet. How many beans are in one row?
Have pairs of students solve this problem, using equipment as required, record what they do and share their methods and answers (solutions)with another pair.

3. Beneath the problem in 2 above, record on the chart in words a student’s method for solving this problem.
For example: First you have to find one fifth so you divide 25 by five. You’re asked for two fifths so you have add two fifths together or times one fifth by two. You write this 25 ÷ 5 = 5, 5 + 5 = 10 or 2 x 5 = 10
Notice which students use equal sharing (into regions) and which use their knowledge of multiplication (and division) to solve the problem. Encourage them to share their ideas.

4. Ensure that equipment is available and pose further problems, having students show and record their solutions in their preferred ways.
For example:
There are 16 beans in a packet. There are 3/8 of the packet in each row. How many is that?
There are 18 beans in a packet. There are 5/9 of the packet in each row. How many is that?
There are 18 beans in a packet. There are 5/6 of the packet.in each row. How many is that?

5. Summarise this with the students. “When you are finding more than a unit fraction of a set, you divide the number in the set by the denominator of the fraction. This gives you the unit fraction of the set. Then you multiply by the numerator of the fraction because this tells you how many of these equal parts are needed.”
For example: To find 3/8 of 16: find 1/8 first by saying 16 ÷ 8 = 2. Then find 3/8 by saying 2 + 2 + 2 = 6 or 3 x 2 = 6. So 6 is 3/8 of 16.

Activity 2

Have students play Telling the Truth (Attachment 5) in pairs.
(Purpose: to identify the correct fractions of sets)
The aim of the game is to be the player to collect the most pairs of questions with correct answers. Five cards are dealt to each player who must firstly decide which of the cards in their hand do not tell the truth. They discard these cards, turning them upside down and placing them to one side (they may need to be checked later in the game). They then find any matching pairs in their hand and place these face up in front of them.
The players then take turns to ask for an answer card to any of the question cards in their hand, or to ask for a question card that matches an answer card in their hand.
Upon Player One’s request for a card, if the Player Two gives an untrue card, Player Two must miss a turn. Player One may immediately make another request.
If Player Two has no suitable cards he tells Player One to pick up from the pile. Each time a player picks up or receives a card they must check it for accuracy.
The player with the most correct matching pairs when all the cards are used, is the winner.

Activity 3

Conclude this session with a discussion of the game and summary of learning.

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